The main aim of this work is to consider a meshfree algorithm for solving Burgers’ equation with the quartic Bspline quasiinterpolation. Quasiinterpolation is very useful in the study of approximation theory and its applications, since it can yield solutions directly without the need to solve any linear system of equations and overcome the illconditioning problem resulting from using the Bspline as a global interpolant. The numerical scheme is presented, by using the derivative of the quasiinterpolation to approximate the spatial derivative of the dependent variable and a low order forward difference to approximate the time derivative of the dependent variable. Compared to other numerical methods, the main advantages of our scheme are higher accuracy and lower computational complexity. Meanwhile, the algorithm is very simple and easy to implement and the numerical experiments show that it is feasible and valid.
Burgers’ equation plays a significant role in various fields, such as turbulence problems, heat conduction, shock waves, continuous stochastic processes, number theory, gas dynamics, and propagation of elastic waves [
Burgers’ equation is a quasilinear parabolic partial differential equation, whose analytic solutions can be constructed from a linear partial differential equation by using HopfCole transformation [
In 1968 Hardy proposed the multiquadric (MQ) which is a kind of radial basis function (RBF). In Franke’s review paper, the MQ was rated as one of the best methods among 29 scattered data interpolation and ease of implementation. Since Kansa successfully applied MQ for solving partial differential equation, more and more reasearchers have been attracted by this meshfree, scattered data approximation scheme [
With the use of univariate multiquadric (MQ) quasiinterpolation, solution of Burgers’ equations was obtained by Chen and Wu [
This paper is arranged as follows. In Section
For an interval
The zero degree Bspline is defined as
We apply this recursion to get the quartic Bspline
In [
We use
Differentiating interpolation polynomials leads to the classic finite difference for the approximate computation of derivatives. Therefore, we can draw a conclusion of approximating derivatives of
By some trivial computations, we can obtain the value of
The values of




Otherwise  






0 





0 





0 





0 
In this section, we present the numerical scheme for solving Burgers’ equation based on the quartic Bspline quasiinterpolation.
Discretizing the Burgers’ equation
We can get
Starting from the initial condition, we can compute the numerical solution of Burgers’ equation step by step using the Bspline quasiinterpolation scheme (
To investigate the applicability of the quasiinterpolation method to Burgers’ equation, four selected example problems are studied. To show the efficiency of the present method for our problem in comparison with the exact solution, we use the following norms to assess the performance of our scheme:
Burgers’ equation is solved over the region
In this computational study, we set
Comparison of exact and numerical solution at





Our method  Asai [ 
Exact  Our method  Asai [ 
Exact  
0.1  0.653563  0.653589  0.653544  0.327870  0.327874  0.327870 
0.2  1.305519  1.305611  1.305534  0.655028  0.655078  0.655069 
0.3  1.949321  1.949485  1.949364  0.978449  0.978427  0.978413 
0.4  2.565977  2.566103  2.565925  1.288417  1.288485  1.288463 
0.5  3.110769  3.110992  3.110739  1.563014  1.563096  1.563064 
0.6  3.492902  3.493222  3.492866  1.756653  1.756691  1.756642 
0.7  3.549538  3.550079  3.549595  1.787184  1.787281  1.787206 
0.8  3.050089  3.050702  3.050134  1.537658  1.537794  1.537694 
0.9  1.816492  1.817077  1.816660  0.916795  0.916941  0.916860 
In this example, we consider the exact solution of Burgers’ equation [
We solve the problem with
The computational results at
BSQI [ 
MQQI [ 
QBCM I [ 
QBGM I [ 
Our method  


3.43253  5.77786  0.77033  1.92558  0.85269 

9.26698  20.8467  3.03817  6.35489  3.79716 
The numerical solutions with
Consider Burgers’ equation with the initial condition
The analytical solution of this problem was given by Cole [
In Table
Comparison of results at different time for


Hassanien [ 
Kutluay [ 
Ozis [ 
Our method  Exact 

0.25  0.4  0.3089  0.3083  0.3143  0.3089  0.3089 
0.6  0.2407  0.2404  0.2437  0.2407  0.2407  
0.8  0.1957  0.1954  0.1976  0.1957  0.1957  
1.0  0.1626  0.1624  0.1639  0.1626  0.1626  
3.0  0.0272  0.0272  0.0274  0.0272  0.0272  


0.50  0.4  0.5696  0.5691  0.5764  0.5696  0.5696 
0.6  0.4472  0.4468  0.4517  0.4472  0.4472  
0.8  0.3592  0.3589  0.3625  0.3592  0.3592  
1.0  0.2919  0.2916  0.2944  0.2919  0.2919  
3.0  0.0402  0.0402  0.0406  0.0402  0.0402  


0.75  0.4  0.6254  0.6256  0.6259  0.6257  0.6254 
0.6  0.4872  0.4570  0.4903  0.4872  0.4872  
0.8  0.3739  0.3737  0.3771  0.3739  0.3739  
1.0  0.2875  0.2872  0.2902  0.2875  0.2875  
3.0  0.0298  0.0297  0.0133  0.0298  0.0298 
Comparison of results at

Hon and Mao [ 
MQQI [ 
BSQI [ 
Our method  Exact 

0.1  0.0664  0.07124  0.06628  0.06630  0.06632 
0.2  0.1313  0.13431  0.13115  0.13119  0.13121 
0.3  0.1928  0.19339  0.19269  0.19271  0.19279 
0.4  0.2481  0.24538  0.24792  0.24797  0.24803 
0.5  0.2919  0.28517  0.29175  0.29185  0.29191 
0.6  0.3159  0.30473  0.31580  0.31598  0.31607 
0.7  0.3079  0.29288  0.30791  0.30800  0.30810 
0.8  0.2534  0.23784  0.25337  0.25344  0.25372 
0.9  0.1459  0.13542  0.14583  0.14587  0.14606 
Comparison of results at

Hon and Mao [ 
MQQI [ 
BSQI [ 
Our method  Exact 

0.1  0.0755  0.07868  0.07530  0.07538  0.0754 
0.2  0.1507  0.15202  0.15049  0.15066  0.1506 
0.3  0.2257  0.22554  0.22544  0.22573  0.2257 
0.4  0.3003  0.29904  0.30002  0.30028  0.3003 
0.5  0.3744  0.37226  0.37407  0.37437  0.3744 
0.6  0.4478  0.44484  0.44742  0.44778  0.4478 
0.7  0.5202  0.51643  0.51985  0.52038  0.5203 
0.8  0.5913  0.58622  0.59106  0.59151  0.5915 
0.9  0.6607  0.62956  0.65964  0.66007  0.6600 
Comparison of results at

Hon and Mao [ 
MQQI [ 
BSQI [ 
Our method 

0.05  0.0422  0.0422  0.0422  0.0422 
0.16  0.1263  0.1263  0.1262  0.1262 
0.27  0.2103  0.2103  0.2096  0.2103 
0.38  0.2939  0.2939  0.2928  0.2939 
0.50  0.3769  0.3769  0.3754  0.3769 
0.61  0.4592  0.4592  0.4573  0.4592 
0.72  0.5404  0.5404  0.5381  0.5404 
0.83  0.6203  0.6201  0.6174  0.6203 
0.94  0.6983  0.6957  0.6947  0.6983 
We consider particular solution of Burgers’ equation:
Comparison of results at different times for















BSQI [ 
1.66464  5.12020  2.06695  6.31491  2.36889  6.85425 
QBCM I [ 
0.19127  0.54058  0.14246  0.39241  0.93617  5.54899 
QBCM II [ 
0.49130  1.16930  0.41864  0.93664  1.28863  7.49147 
MQQI [ 
6.88480  25.6767  7.89738  27.2424  8.56856  2.68122 
Our method  0.6642  0.91725  0.7573  1.1465  0.8592  1.2103 







BSQI [ 
0.82751  2.59444  0.98595  2.35031  1.58264  5.73827 
QBCM I [ 
0.17014  0.40431  0.20476  0.86363  1.29951  6.69425 
QBCM II [ 
0.24003  0.48800  0.30849  1.14760  1.57548  8.06798 
MQQI [ 
5.89555  14.7550  6.64358  15.9892  6.90385  16.3403 
Our method  0.82751  0.50367  0.46281  1.05625  0.88261  4.73827 
The numerical solutions for
Following the recent development of the quasiinterpolation method for scattered data interpolation and the meshfree method for solving partial differential equations, this paper combines these ideas and proposes a new meshfree quasiinterpolation method for Burgers’ equation. The method does not require solving a large size matrix equation and, hence, the illconditioning problem from using Bspline functions as global interpolants can be avoided. We have made comparison studies between the present results and the exact solutions. The agreement of our numerical results with those exact solutions is excellent. For the highdimensional Burgers’ equations, we believe our scheme can also be applicable. In this case, we would use multivariate spline quasiinterpolation instead of univariate spline quasiinterpolation. We will consider these problems in our future work.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is supported by the Disciplinary Construction Guide Foundation of Harbin Institute of Technology at Weihai (no. WH20140206) and the Scientific Research Foundation of Harbin Institute of Technology at Weihai (no. HIT(WH)201319).